Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.420=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Afghanistan	82% (64%-97%)	15,750	265
Albania	87% (45%-100%)	1,143	33
Algeria	33% (23%-46%)	9,513	661
Andorra	23% (11%-44%)	765	51
Argentina	50% (38%-63%)	17,402	556
Armenia	60% (43%-79%)	9,492	139
Australia	87% (56%-100%)	7,204	103
Austria	26% (13%-46%)	16,663	668
Azerbaijan	82% (58%-100%)	5,662	68
Bahamas	59% (17%-100%)	102	11
Bahrain	98% (90%-100%)	11,804	19
Bangladesh	78% (60%-100%)	49,534	672
Belarus	99% (91%-100%)	43,403	240
Belgium	27% (21%-33%)	58,517	9,486
Bolivia	33% (25%-42%)	10,531	343
Bosnia and Herzegovina	35% (16%-66%)	2,523	153
Brazil	26% (21%-31%)	526,447	29,937
Bulgaria	23% (16%-32%)	2,513	144
Burkina Faso	42% (20%-81%)	881	53
Cameroon	46% (25%-87%)	6,397	199
Canada	14% (11%-17%)	91,694	7,326
Chad	51% (27%-81%)	790	66
Chile	82% (66%-98%)	105,159	1,113
China	91% (26%-100%)	84,154	4,638
Colombia	36% (28%-45%)	30,493	969
Congo	50% (25%-90%)	611	20
Costa Rica	78% (40%-100%)	1,084	10
Cote dIvoire	90% (67%-100%)	2,951	33
Croatia	24% (12%-45%)	2,246	103
Cuba	65% (34%-97%)	2,083	83
Cyprus	85% (52%-100%)	949	17
Czechia	44% (31%-64%)	9,302	321
Democratic Republic of the Congo	84% (55%-100%)	3,194	72
Denmark	59% (39%-82%)	11,699	576
Djibouti	83% (55%-100%)	3,569	24
Dominican Republic	84% (67%-97%)	17,572	502
Ecuador	30% (24%-37%)	39,994	3,394
Egypt	27% (21%-34%)	26,384	1,005
El Salvador	62% (41%-91%)	2,582	46
Equatorial Guinea	83% (52%-100%)	1,306	12
Estonia	34% (20%-55%)	1,870	68
Finland	60% (33%-90%)	6,885	320
France	22% (17%-26%)	152,091	28,833
Gabon	94% (77%-100%)	2,655	17
Georgia	78% (41%-100%)	796	12
Germany	40% (31%-50%)	182,028	8,522
Ghana	98% (89%-100%)	8,070	36
Greece	25% (15%-39%)	2,917	175
Guatemala	36% (26%-48%)	5,336	116
Guernsey	46% (14%-96%)	252	13
Guinea	95% (80%-100%)	3,844	23
Guyana	46% (13%-97%)	153	12
Haiti	55% (31%-86%)	2,226	45
Honduras	43% (30%-59%)	5,362	217
Hungary	11% (7.3%-16%)	3,921	532
Iceland	87% (53%-100%)	1,806	10
India	35% (29%-42%)	198,706	5,598
Indonesia	21% (17%-26%)	26,940	1,641
Iran	51% (41%-62%)	154,445	7,878
Iraq	27% (19%-38%)	6,868	215
Ireland	24% (15%-35%)	25,062	1,650
Isle of Man	37% (9.6%-97%)	336	24
Israel	80% (57%-99%)	17,219	287
Italy	14% (12%-17%)	233,197	33,475
Japan	15% (11%-21%)	16,930	894
Jersey	17% (6.8%-46%)	308	29
Kazakhstan	98% (90%-100%)	11,571	41
Kenya	32% (20%-53%)	2,021	69
Kosovo	66% (37%-99%)	1,064	30
Kuwait	96% (84%-100%)	27,762	220
Kyrgyzstan	88% (62%-100%)	1,845	17
Latvia	49% (25%-87%)	1,071	24
Lebanon	81% (47%-100%)	1,233	27
Liberia	26% (9.7%-71%)	296	27
Lithuania	24% (11%-39%)	1,678	70
Luxembourg	51% (32%-74%)	4,019	110
Malaysia	97% (82%-100%)	7,857	115
Mali	20% (13%-28%)	1,315	78
Mauritius	62% (18%-100%)	335	10
Mexico	13% (10%-15%)	93,435	10,167
Moldova	30% (23%-39%)	8,360	307
Morocco	98% (83%-100%)	7,833	205
Netherlands	22% (17%-28%)	46,545	5,962
New Zealand	49% (21%-91%)	1,154	22
Nicaragua	81% (48%-100%)	759	35
Niger	19% (9.3%-35%)	958	65
Nigeria	44% (30%-60%)	10,578	299
North Macedonia	13% (7.3%-21%)	2,315	140
Norway	82% (50%-100%)	8,411	236
Oman	97% (88%-100%)	12,223	50
Pakistan	34% (27%-43%)	76,398	1,621
Panama	59% (43%-76%)	13,837	344
Paraguay	92% (68%-100%)	995	11
Peru	41% (31%-49%)	170,039	4,634
Philippines	40% (28%-53%)	18,638	960
Poland	56% (43%-70%)	24,165	1,074
Portugal	26% (19%-34%)	32,700	1,424
Puerto Rico	74% (50%-96%)	3,873	136
Qatar	90% (53%-100%)	58,433	40
Romania	25% (19%-32%)	19,398	1,279
Russia	37% (31%-45%)	423,741	5,037
San Marino	83% (35%-100%)	687	42
Saudi Arabia	87% (54%-100%)	87,142	525
Senegal	87% (64%-100%)	3,739	42
Serbia	94% (73%-100%)	11,430	244
Sierra Leone	40% (20%-69%)	861	46
Singapore	90% (56%-100%)	35,292	24
Sint Maarten	12% (4.1%-33%)	77	15
Slovakia	73% (43%-99%)	1,522	28
Slovenia	18% (9.9%-32%)	1,483	108
Somalia	40% (19%-70%)	2,023	79
South Africa	40% (32%-48%)	34,357	705
South Korea	60% (30%-96%)	11,541	272
Spain	9.4% (7.7%-11%)	239,638	27,940
Sudan	9.3% (6.9%-12%)	5,173	298
Sweden	19% (15%-24%)	37,814	4,403
Switzerland	30% (21%-41%)	30,788	1,656
Tajikistan	97% (86%-100%)	3,930	47
Thailand	78% (49%-100%)	3,082	57
Togo	75% (37%-100%)	443	13
Tunisia	52% (21%-98%)	1,084	48
Turkey	77% (62%-91%)	164,769	4,563
Ukraine	42% (31%-55%)	24,340	727
United Arab Emirates	99% (92%-100%)	35,192	266
United Kingdom	14% (11%-16%)	276,332	39,045
United Republic of Tanzania	57% (28%-96%)	509	21
United States of America	35% (29%-41%)	1,811,277	105,147
Uruguay	45% (22%-83%)	825	23
Uzbekistan	95% (80%-100%)	3,718	15
Venezuela	88% (61%-100%)	1,662	17
Yemen	3.2% (2.2%-4.5%)	354	84

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

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2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim